For finite groups number of inequivalent irreducible complex representations equals the number of conjugacy classes.
When the group is $S_n$ it is easy to see that both are available one each for every partition of $n$, into positive integers. So there is a direct correspondence between the two collections associated to the group.
For cyclic groups of order $n$, conjugacy classes are singletons, and hence for the conjugacy class of $g^k$ (g being a fixed generator of the group) one can associate the representation $g\mapsto e^{2\pi i k/n}$.
Is there any other group admitting similar EXPLICIT CORRESPONDENCE between irreducible representations and conjugacy classes?